458 



NA TURE 



[March 20, 1902 



The contention that Euclid's " Elements " is an unsuit- 

 able text-book is borne out by a critical scrutiny. The 

 book was an attempt to deduce a complete system of 

 geometry from a minimum number of assumptions, con- 

 tained in definitions, axioms and postulates. No such 

 attempt, even if completely successful, can possibly ap- 

 peal to immature minds ; and it is well known that 

 Euclid's attempt, in spite of its many conspicuous merits, 

 is logically defective. This would not matter very much 

 if it were otherwise well adapted to effect the two objects 

 which we have noted above as the true objects to 

 be aimed at in teaching geometry. Notoriously it is 

 not so adapted ; striking instances of its deadening effect 

 upon the minds of average boys have been recorded by 

 Prof. Minchin and others. Such instances may fairly be 

 cited in opposition to the claim, urged by the supporters 

 of Euclid, that his system affords valuable mental train- 

 ing ; and this clami does not gain in force when it is 

 observed that most of the men who have undergone the 

 training appear to be unable to appreciate the logical 

 defects of the system, and that they accept without ques- 

 tion the absurdities that are often made to do duty, in our 

 text-books of analysis and mechanics, as definitions or 

 as proofs. Further, the tacit agreement to drop Euclid's 

 fifth book has robbed his sequence of its significance. 

 This book is really the great contribution of antiquity to 

 the problem of irrational numbers — perhaps the central 

 problem of mathematics. It is certainly the keystone of 

 Euclid's system. Without it, there is no reason why pro- 

 portion and measurement, treated arithmetically, should 

 not take a much earlier place in a system of geometry. 

 In elementary stages, it will probably be best to postpone 

 all questions of incommensurable magnitudes and irra- 

 tional numbers ; but sooner or later, m the training of 

 a mathematician, such questions must be faced. The 

 Greeks approached arithmetic through geometry, and for 

 two thousand years Euclid's fifth book was the only theory 

 of irrational nnmbers. Now, however, the Greek way is 

 not the only way, nor, as 1 believe, the most excellent. 

 There exists now a complete arithmetical theory of 

 irrational numbers ; and the theory of exact measurement 

 of incommensurable, as well as of commensurable, mag- 

 nitudes can be founded on a secure arithmetical basis. 

 This revolution of mathematical thought was accom- 

 plished in Germany in the second half of the nineteenth 

 century ; its full effect has yet to be felt. 



But geometry is not the only branch, even of elementary 

 mathematics, of which the aspect would be changed en- 

 tirely by giving effect to the above-stated principles of 

 reform ; nor is the need for improvement confined to the 

 teaching of elementary mathematics. In algebra, for 

 example, the changes advocated in the book under 

 notice, or those proposed in the memorial of some school- 

 masters already referred to, would constitute great im- 

 provements in the method of presenting the subject. 

 Here again the P'rench text-books are much better than 

 most of ours. In analysis generally, the traditional order 

 of the topics stands in need of drastic alteration — boys 

 and girls ought not to be taught to expand the circular 

 functions in infinite products before they have ever 

 plotted a graph or differentiated the simplest expression. 

 Prof. Perry has been a constant advocate of graphic 

 NO. 1690, VOL. 65] 



methods and of the early study of the differential and inte- 

 gral calculus. He holds that this study should precede that 

 of many things which now come before it, e.g. advanced 

 analytical geometry of conies ; this view is put forward 

 on the ground that such an order would be more helpful 

 to those students who will afterwards require to use 

 mathematics in the practice of their professions. They 

 are not the only students who would gain by the change. 

 The educational value of mathematics would be increased 

 enormously. Mathematics becomes an instrument of 

 liberal education, not merely by the practice of processes 

 — they are means to an end — nor yet by the storing of 

 information, though knowledge of facts is an element of 

 culture — but by the formation of exact ideas as regards 

 both the definiteness of its fundamental notions and the 

 inevitableness of its results. The impression that is 

 made upon the mind, when one realises the inexorable 

 necessity of the conclusion of a chain of reasoning, is the 

 element in mathematical training that has been empha- 

 sized the most by those who support the claim of mathe- 

 matics to be considered an integral part of a liberal 

 education ; but it may be held that the illumination to be 

 derived from any of the fundamental notions of mathe- 

 matics — such notions, for example, as proportion, con- 

 tinuity, vector, group — when they are thoroughly grasped, 

 is a not less important element from the same point of 

 view. The traditional order of study has tended to 

 obscure the fundamental notions and the general drift of 

 the arguments under a cloud of secondary developments. 

 That the differential and integral calculus can be pre- 

 sented at such a stage as that indicated by Prof Perry, 

 in a manner so practical as to suit the student of 

 engineering, and at the same time so rigorous and so 

 luminous as to be a worthy means of liberal education, 

 has been shown by more than one recent treatise. The 

 examinations that have most influence upon the order of 

 study near this stage are probably those for entrance 

 scholarships at the colleges of the older Universities. 

 A few reforming tutors might now initiate a change 

 that would produce a very great effect. 



It is unnecessary here to follow out the application of 

 the principles of reform to mechanics, or to the remain- 

 ing subjects of a school course of mathematics. The 

 book under notice contains numerous suggestions on these 

 heads. Some readers may be inclined to think that in 

 the introductory address, and in some of the subsequent 

 speeches, undue prominence was given to the needs of 

 students who are destined to become engineers, or 

 teachers in primary schools. A view that is held widely, 

 but has not perhaps been emphasized sufficiently, is that, 

 in the stage of ordinary school work, the course that is 

 most suitable for such classes of students is also precisely 

 the best for those for whom mathematics is meant to be a 

 means of culture and for those who have the ability, and 

 will afterwards have opportunities, to assist in the de- 

 velopment of mathematical theories. To bring such a 

 course into general use will require much persistent effort, 

 directed continually to one end ; and the first step will 

 necessarily be the conversion of examiners and of the 

 bodies that make regulations for the conduct of examina- 

 tions. The future of mathematical teaching in this 

 country is in their hands. .■\. E. H. L. 



